Optimal. Leaf size=35 \[ \frac{1}{2} \cot (x) \sqrt{-\csc ^2(x)}-\frac{1}{2} \tan ^{-1}\left (\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right ) \]
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Rubi [A] time = 0.0255565, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3657, 4122, 195, 217, 203} \[ \frac{1}{2} \cot (x) \sqrt{-\csc ^2(x)}-\frac{1}{2} \tan ^{-1}\left (\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right ) \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4122
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \left (-1-\cot ^2(x)\right )^{3/2} \, dx &=\int \left (-\csc ^2(x)\right )^{3/2} \, dx\\ &=\operatorname{Subst}\left (\int \sqrt{-1-x^2} \, dx,x,\cot (x)\right )\\ &=\frac{1}{2} \cot (x) \sqrt{-\csc ^2(x)}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x^2}} \, dx,x,\cot (x)\right )\\ &=\frac{1}{2} \cot (x) \sqrt{-\csc ^2(x)}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right )\\ &=-\frac{1}{2} \tan ^{-1}\left (\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right )+\frac{1}{2} \cot (x) \sqrt{-\csc ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.0796303, size = 48, normalized size = 1.37 \[ -\frac{\csc \left (\frac{x}{2}\right ) \sec \left (\frac{x}{2}\right ) \left (-\log \left (\sin \left (\frac{x}{2}\right )\right )+\log \left (\cos \left (\frac{x}{2}\right )\right )+\cot (x) \csc (x)\right )}{4 \sqrt{-\csc ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 32, normalized size = 0.9 \begin{align*}{\frac{\cot \left ( x \right ) }{2}\sqrt{-1- \left ( \cot \left ( x \right ) \right ) ^{2}}}-{\frac{1}{2}\arctan \left ({\cot \left ( x \right ){\frac{1}{\sqrt{-1- \left ( \cot \left ( x \right ) \right ) ^{2}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.6474, size = 383, normalized size = 10.94 \begin{align*} \frac{{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) -{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + 2 \,{\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - 2 \,{\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \sin \left (4 \, x\right ) - 2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + 4 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \cos \left (x\right ) \sin \left (2 \, x\right ) - 4 \, \cos \left (2 \, x\right ) \sin \left (x\right ) + 2 \, \sin \left (x\right )}{2 \,{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.7396, size = 232, normalized size = 6.63 \begin{align*} \frac{{\left (-i \, e^{\left (4 i \, x\right )} + 2 i \, e^{\left (2 i \, x\right )} - i\right )} \log \left (e^{\left (i \, x\right )} + 1\right ) +{\left (i \, e^{\left (4 i \, x\right )} - 2 i \, e^{\left (2 i \, x\right )} + i\right )} \log \left (e^{\left (i \, x\right )} - 1\right ) + 2 i \, e^{\left (3 i \, x\right )} + 2 i \, e^{\left (i \, x\right )}}{2 \,{\left (e^{\left (4 i \, x\right )} - 2 \, e^{\left (2 i \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- \cot ^{2}{\left (x \right )} - 1\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.15838, size = 46, normalized size = 1.31 \begin{align*} -\frac{1}{4} \,{\left (\frac{2 i \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 1} - i \, \log \left (\cos \left (x\right ) + 1\right ) + i \, \log \left (-\cos \left (x\right ) + 1\right )\right )} \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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